Two-layer neural networks with values in a Banach space
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We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of ℝ^d-valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates, extending existing results for the finite-dimensional case. In the second part of the paper, we consider training such networks using a finite amount of noisy observations from the regularisation theory viewpoint. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates.
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